President Obama and House Speaker John A. Boehner are continuing their talks, but key officials in both parties now believe that Washington will be unable to avoid some mix of the tax increases and automatic spending cuts mandated next month by the austerity measures known as the “fiscal cliff.”

Senior Democratic and Republican officials say the best-case scenario will be for a less ambitious deal to extend middle-class tax cuts and forestall tax hikes on most Americans.

But such a deal would still set in motion a series of steep spending cuts at some federal agencies and allow key tax provisions to expire, raising taxes for many. “I don’t know if we fall off the cliff, but I think we’re at least going to jump out of a tree,” said Sen. Lindsey O. Graham (R-S.C.).

That assessment hasn’t stopped Obama and Boehner (R-Ohio) from trying to reach a broader pact before year’s end. The two spoke by phone Friday afternoon, after news broke of the massacre at an elementary school in Connecticut, and officials in both parties confirmed that Boehner was considering allowing tax rates to rise for those earning more than $1 million annually, in exchange for cuts in entitlement programs.

That would represent a significant concession for Boehner, who has opposed any increase in rates and warned that such a plan could not pass the GOP-dominated House.

But the White House, which wants higher rates on annual incomes above $250,000, rejected Boehner’s offer, according to officials familiar with the talks.

Obama, Democrats says, believes he has already won on this issue, having won reelection by campaigning on a pledge to raise taxes on the wealthy, which polls show that a majority supports.

Instead, congressional Democrats say, Obama is only prepared to discuss cuts to Medicare and Social Security if Republicans are willing to reform the process for lifting the Treasury’s borrowing authority, surrendering the GOP’s ability to use a debt-ceiling fight early next year as a leverage point for more spending cuts.

This standoff has left many leaders in both parties searching for fallback plans.

White House press secretary Jay Carney told reporters Thursday, “In case of emergency, the House should break the glass; the House speaker ought to allow the Republicans to vote on extending tax cuts for 98 percent of the American people. That would deal with a chunk of the so-called fiscal cliff.”

The hope is to blunt the impact on the economy of the more than $500 billion in annual automatic tax increases and spending cuts scheduled to take effect in January, and buy time into next year for a broader deal involving comprehensive reforms of the tax code and federal entitlement programs.

No new face-to-face negotiations are scheduled for Obama and Boehner. The speaker’s aides declined to comment on the issue of Boehner offering a tax hike on millionaires, which was first reported by Politico. “We have not commented on the content of private discussions, and we’re certainly not going to comment on rumors,” Michael Steel, Boehner’s spokesman, said in a statement.